Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.7% → 99.7%

Time: 18.3s

Alternatives: 13

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 1e+307) t_1 U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+307) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+307) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+307:
		tmp = t_1
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+307)
		tmp = t_1;
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+307)
		tmp = t_1;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+307], t$95$1, U]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -inf.0

   1. Initial program 5.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 5.7%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 5.7%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 5.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 55.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 55.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 55.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 55.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 48.0%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 48.0%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 48.0%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 9.99999999999999986e306

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 4.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 4.8%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 4.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 4.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 57.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 57.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 57.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 57.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 47.8%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+307}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 57.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t_0\\ \mathbf{if}\;K \leq 62000000:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;K \leq 5.3 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 1.22 \cdot 10^{+240}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot t_0}, t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (* (* -2.0 J) t_0)))
   (if (<= K 62000000.0)
     (*
      (hypot 1.0 (/ U (* J (* 2.0 (cos (/ K 2.0))))))
      (* J (+ -2.0 (* 0.25 (* K K)))))
     (if (<= K 5.3e+213)
       t_1
       (if (<= K 1.22e+240)
         (* -2.0 (+ (* J (/ J U)) (* U 0.5)))
         (fma -0.25 (/ (* U U) (* J t_0)) t_1))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = (-2.0 * J) * t_0;
	double tmp;
	if (K <= 62000000.0) {
		tmp = hypot(1.0, (U / (J * (2.0 * cos((K / 2.0)))))) * (J * (-2.0 + (0.25 * (K * K))));
	} else if (K <= 5.3e+213) {
		tmp = t_1;
	} else if (K <= 1.22e+240) {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	} else {
		tmp = fma(-0.25, ((U * U) / (J * t_0)), t_1);
	}
	return tmp;
}

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	tmp = 0.0
	if (K <= 62000000.0)
		tmp = Float64(hypot(1.0, Float64(U / Float64(J * Float64(2.0 * cos(Float64(K / 2.0)))))) * Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K)))));
	elseif (K <= 5.3e+213)
		tmp = t_1;
	elseif (K <= 1.22e+240)
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	else
		tmp = fma(-0.25, Float64(Float64(U * U) / Float64(J * t_0)), t_1);
	end
	return tmp
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[K, 62000000.0], N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[K, 5.3e+213], t$95$1, If[LessEqual[K, 1.22e+240], N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(U * U), $MachinePrecision] / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if K < 6.2e7

   1. Initial program 71.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 71.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 71.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 58.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 58.4%

         \[\leadsto \left(\color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

      2. distribute-rgt-out 58.4%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.25 \cdot {K}^{2} + -2\right)\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

      3. unpow2 58.4%

         \[\leadsto \left(J \cdot \left(0.25 \cdot \color{blue}{\left(K \cdot K\right)} + -2\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

   6. Simplified 58.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.25 \cdot \left(K \cdot K\right) + -2\right)\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

   if 6.2e7 < K < 5.2999999999999998e213

   1. Initial program 67.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 67.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 67.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 80.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 80.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 80.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 45.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 45.0%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 45.0%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 45.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 45.0%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 45.0%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 45.0%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   if 5.2999999999999998e213 < K < 1.2200000000000001e240

   1. Initial program 35.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 35.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 35.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 35.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 68.4%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 68.4%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 68.4%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 68.4%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 69.3%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 69.3%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 69.3%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]

   if 1.2200000000000001e240 < K

   1. Initial program 67.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 67.2%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 67.2%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 67.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 89.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 89.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 89.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 34.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{{U}^{2}}{\cos \left(0.5 \cdot K\right) \cdot J} + -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. fma-def 34.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{{U}^{2}}{\cos \left(0.5 \cdot K\right) \cdot J}, -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} \]

      2. unpow2 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{U \cdot U}}{\cos \left(0.5 \cdot K\right) \cdot J}, -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) \]

      3. *-commutative 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{\color{blue}{J \cdot \cos \left(0.5 \cdot K\right)}}, -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) \]

      4. associate-*r* 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J}\right) \]

      5. *-commutative 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J\right) \]

      6. *-commutative 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J\right) \]

      7. associate-*r* 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)}\right) \]

      8. *-commutative 34.6%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right)\right) \]

   6. Simplified 34.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}, \cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 62000000:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;K \leq 5.3 \cdot 10^{+213}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;K \leq 1.22 \cdot 10^{+240}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)}, \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(J \cdot \left(-2 \cdot t_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right) \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* J (* -2.0 t_0)) (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return (J * (-2.0 * t_0)) * Math.hypot(1.0, (U / (J * (2.0 * t_0))));
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return (J * (-2.0 * t_0)) * math.hypot(1.0, (U / (J * (2.0 * t_0))))

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(J * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.3%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Step-by-step derivation

   1. *-commutative 70.3%

      \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. associate-*l* 70.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. unpow2 70.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

   4. hypot-1-def 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   5. *-commutative 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   6. associate-*l* 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Simplified 86.3%

   \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

4. Final simplification 86.3%

   \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

Alternative 4: 57.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;K \leq 62000000:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;K \leq 6.5 \cdot 10^{+213} \lor \neg \left(K \leq 9.5 \cdot 10^{+239}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= K 62000000.0)
   (*
    (hypot 1.0 (/ U (* J (* 2.0 (cos (/ K 2.0))))))
    (* J (+ -2.0 (* 0.25 (* K K)))))
   (if (or (<= K 6.5e+213) (not (<= K 9.5e+239)))
     (* (* -2.0 J) (cos (* K 0.5)))
     (* -2.0 (+ (* J (/ J U)) (* U 0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (K <= 62000000.0) {
		tmp = hypot(1.0, (U / (J * (2.0 * cos((K / 2.0)))))) * (J * (-2.0 + (0.25 * (K * K))));
	} else if ((K <= 6.5e+213) || !(K <= 9.5e+239)) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (K <= 62000000.0) {
		tmp = Math.hypot(1.0, (U / (J * (2.0 * Math.cos((K / 2.0)))))) * (J * (-2.0 + (0.25 * (K * K))));
	} else if ((K <= 6.5e+213) || !(K <= 9.5e+239)) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if K <= 62000000.0:
		tmp = math.hypot(1.0, (U / (J * (2.0 * math.cos((K / 2.0)))))) * (J * (-2.0 + (0.25 * (K * K))))
	elif (K <= 6.5e+213) or not (K <= 9.5e+239):
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	else:
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (K <= 62000000.0)
		tmp = Float64(hypot(1.0, Float64(U / Float64(J * Float64(2.0 * cos(Float64(K / 2.0)))))) * Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K)))));
	elseif ((K <= 6.5e+213) || !(K <= 9.5e+239))
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (K <= 62000000.0)
		tmp = hypot(1.0, (U / (J * (2.0 * cos((K / 2.0)))))) * (J * (-2.0 + (0.25 * (K * K))));
	elseif ((K <= 6.5e+213) || ~((K <= 9.5e+239)))
		tmp = (-2.0 * J) * cos((K * 0.5));
	else
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[K, 62000000.0], N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[K, 6.5e+213], N[Not[LessEqual[K, 9.5e+239]], $MachinePrecision]], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if K < 6.2e7

   1. Initial program 71.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 71.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 71.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 86.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 58.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 58.4%

         \[\leadsto \left(\color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

      2. distribute-rgt-out 58.4%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.25 \cdot {K}^{2} + -2\right)\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

      3. unpow2 58.4%

         \[\leadsto \left(J \cdot \left(0.25 \cdot \color{blue}{\left(K \cdot K\right)} + -2\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

   6. Simplified 58.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.25 \cdot \left(K \cdot K\right) + -2\right)\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \]

   if 6.2e7 < K < 6.49999999999999982e213 or 9.5000000000000008e239 < K

   1. Initial program 67.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 67.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 67.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 83.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 83.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 83.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 41.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 41.4%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 41.4%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 41.4%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 41.4%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 41.4%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 41.4%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   if 6.49999999999999982e213 < K < 9.5000000000000008e239

   1. Initial program 35.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 35.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 35.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 35.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 68.4%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 68.4%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 68.4%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 68.4%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 69.3%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 69.3%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 69.3%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 62000000:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right) \cdot \left(J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{elif}\;K \leq 6.5 \cdot 10^{+213} \lor \neg \left(K \leq 9.5 \cdot 10^{+239}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 65.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -8 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -7.8 \cdot 10^{-110}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.05 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-120}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.6 \cdot 10^{-89} \lor \neg \left(J \leq 12500000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= J -8e+16)
     t_0
     (if (<= J -8.5e-53)
       U
       (if (<= J -3.3e-92)
         t_0
         (if (<= J -7.8e-110)
           (- U)
           (if (<= J -1e-310)
             U
             (if (<= J 4.05e-243)
               (- U)
               (if (<= J 2.1e-229)
                 U
                 (if (<= J 5e-120)
                   (- U)
                   (if (or (<= J 6.6e-89) (not (<= J 12500000000000.0)))
                     t_0
                     (* -2.0 (+ (* J (/ J U)) (* U 0.5))))))))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (J <= -8e+16) {
		tmp = t_0;
	} else if (J <= -8.5e-53) {
		tmp = U;
	} else if (J <= -3.3e-92) {
		tmp = t_0;
	} else if (J <= -7.8e-110) {
		tmp = -U;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 4.05e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 5e-120) {
		tmp = -U;
	} else if ((J <= 6.6e-89) || !(J <= 12500000000000.0)) {
		tmp = t_0;
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * j) * cos((k * 0.5d0))
    if (j <= (-8d+16)) then
        tmp = t_0
    else if (j <= (-8.5d-53)) then
        tmp = u
    else if (j <= (-3.3d-92)) then
        tmp = t_0
    else if (j <= (-7.8d-110)) then
        tmp = -u
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 4.05d-243) then
        tmp = -u
    else if (j <= 2.1d-229) then
        tmp = u
    else if (j <= 5d-120) then
        tmp = -u
    else if ((j <= 6.6d-89) .or. (.not. (j <= 12500000000000.0d0))) then
        tmp = t_0
    else
        tmp = (-2.0d0) * ((j * (j / u)) + (u * 0.5d0))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * Math.cos((K * 0.5));
	double tmp;
	if (J <= -8e+16) {
		tmp = t_0;
	} else if (J <= -8.5e-53) {
		tmp = U;
	} else if (J <= -3.3e-92) {
		tmp = t_0;
	} else if (J <= -7.8e-110) {
		tmp = -U;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 4.05e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 5e-120) {
		tmp = -U;
	} else if ((J <= 6.6e-89) || !(J <= 12500000000000.0)) {
		tmp = t_0;
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = (-2.0 * J) * math.cos((K * 0.5))
	tmp = 0
	if J <= -8e+16:
		tmp = t_0
	elif J <= -8.5e-53:
		tmp = U
	elif J <= -3.3e-92:
		tmp = t_0
	elif J <= -7.8e-110:
		tmp = -U
	elif J <= -1e-310:
		tmp = U
	elif J <= 4.05e-243:
		tmp = -U
	elif J <= 2.1e-229:
		tmp = U
	elif J <= 5e-120:
		tmp = -U
	elif (J <= 6.6e-89) or not (J <= 12500000000000.0):
		tmp = t_0
	else:
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J <= -8e+16)
		tmp = t_0;
	elseif (J <= -8.5e-53)
		tmp = U;
	elseif (J <= -3.3e-92)
		tmp = t_0;
	elseif (J <= -7.8e-110)
		tmp = Float64(-U);
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 4.05e-243)
		tmp = Float64(-U);
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 5e-120)
		tmp = Float64(-U);
	elseif ((J <= 6.6e-89) || !(J <= 12500000000000.0))
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = (-2.0 * J) * cos((K * 0.5));
	tmp = 0.0;
	if (J <= -8e+16)
		tmp = t_0;
	elseif (J <= -8.5e-53)
		tmp = U;
	elseif (J <= -3.3e-92)
		tmp = t_0;
	elseif (J <= -7.8e-110)
		tmp = -U;
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 4.05e-243)
		tmp = -U;
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 5e-120)
		tmp = -U;
	elseif ((J <= 6.6e-89) || ~((J <= 12500000000000.0)))
		tmp = t_0;
	else
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -8e+16], t$95$0, If[LessEqual[J, -8.5e-53], U, If[LessEqual[J, -3.3e-92], t$95$0, If[LessEqual[J, -7.8e-110], (-U), If[LessEqual[J, -1e-310], U, If[LessEqual[J, 4.05e-243], (-U), If[LessEqual[J, 2.1e-229], U, If[LessEqual[J, 5e-120], (-U), If[Or[LessEqual[J, 6.6e-89], N[Not[LessEqual[J, 12500000000000.0]], $MachinePrecision]], t$95$0, N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -8e16 or -8.50000000000000044e-53 < J < -3.29999999999999998e-92 or 5.00000000000000007e-120 < J < 6.5999999999999993e-89 or 1.25e13 < J

   1. Initial program 93.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 93.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 93.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 93.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 73.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 73.4%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 73.4%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 73.4%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 73.4%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 73.4%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 73.4%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   if -8e16 < J < -8.50000000000000044e-53 or -7.8e-110 < J < -9.999999999999969e-311 or 4.0499999999999999e-243 < J < 2.09999999999999983e-229

   1. Initial program 37.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 37.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 37.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 40.8%

      \[\leadsto \color{blue}{U} \]

   if -3.29999999999999998e-92 < J < -7.8e-110 or -9.999999999999969e-311 < J < 4.0499999999999999e-243 or 2.09999999999999983e-229 < J < 5.00000000000000007e-120

   1. Initial program 47.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 47.3%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 47.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 66.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 66.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 66.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 66.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 41.5%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 41.5%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 41.5%

      \[\leadsto \color{blue}{-U} \]

   if 6.5999999999999993e-89 < J < 1.25e13

   1. Initial program 57.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 57.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 57.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 57.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 48.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 48.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 48.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 48.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 48.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 32.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 32.1%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 32.1%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 32.1%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 32.1%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 32.1%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 32.1%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 32.1%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 32.1%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 32.1%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+16}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -7.8 \cdot 10^{-110}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.05 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-120}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.6 \cdot 10^{-89} \lor \neg \left(J \leq 12500000000000\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 65.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -5.2 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.5 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -8.8 \cdot 10^{-110}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.35 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.6 \cdot 10^{-120}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{-91} \lor \neg \left(J \leq 1.95 \cdot 10^{+25}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= J -5.2e+16)
     t_0
     (if (<= J -3.9e-53)
       U
       (if (<= J -6.5e-90)
         t_0
         (if (<= J -8.8e-110)
           (- U)
           (if (<= J -1e-310)
             U
             (if (<= J 5.8e-243)
               (- U)
               (if (<= J 2.35e-229)
                 U
                 (if (<= J 5.6e-120)
                   (* -2.0 (fma U 0.5 (/ (* J J) U)))
                   (if (or (<= J 1.6e-91) (not (<= J 1.95e+25)))
                     t_0
                     (* -2.0 (+ (* J (/ J U)) (* U 0.5))))))))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (J <= -5.2e+16) {
		tmp = t_0;
	} else if (J <= -3.9e-53) {
		tmp = U;
	} else if (J <= -6.5e-90) {
		tmp = t_0;
	} else if (J <= -8.8e-110) {
		tmp = -U;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 2.35e-229) {
		tmp = U;
	} else if (J <= 5.6e-120) {
		tmp = -2.0 * fma(U, 0.5, ((J * J) / U));
	} else if ((J <= 1.6e-91) || !(J <= 1.95e+25)) {
		tmp = t_0;
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J <= -5.2e+16)
		tmp = t_0;
	elseif (J <= -3.9e-53)
		tmp = U;
	elseif (J <= -6.5e-90)
		tmp = t_0;
	elseif (J <= -8.8e-110)
		tmp = Float64(-U);
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = Float64(-U);
	elseif (J <= 2.35e-229)
		tmp = U;
	elseif (J <= 5.6e-120)
		tmp = Float64(-2.0 * fma(U, 0.5, Float64(Float64(J * J) / U)));
	elseif ((J <= 1.6e-91) || !(J <= 1.95e+25))
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	end
	return tmp
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -5.2e+16], t$95$0, If[LessEqual[J, -3.9e-53], U, If[LessEqual[J, -6.5e-90], t$95$0, If[LessEqual[J, -8.8e-110], (-U), If[LessEqual[J, -1e-310], U, If[LessEqual[J, 5.8e-243], (-U), If[LessEqual[J, 2.35e-229], U, If[LessEqual[J, 5.6e-120], N[(-2.0 * N[(U * 0.5 + N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[J, 1.6e-91], N[Not[LessEqual[J, 1.95e+25]], $MachinePrecision]], t$95$0, N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -5.2e16 or -3.9000000000000002e-53 < J < -6.4999999999999996e-90 or 5.59999999999999988e-120 < J < 1.59999999999999998e-91 or 1.9500000000000001e25 < J

   1. Initial program 93.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 93.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 93.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 93.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 98.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 73.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 73.4%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 73.4%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 73.4%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 73.4%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 73.4%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 73.4%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   if -5.2e16 < J < -3.9000000000000002e-53 or -8.7999999999999997e-110 < J < -9.999999999999969e-311 or 5.79999999999999953e-243 < J < 2.35000000000000017e-229

   1. Initial program 37.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 37.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 37.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 71.3%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 40.8%

      \[\leadsto \color{blue}{U} \]

   if -6.4999999999999996e-90 < J < -8.7999999999999997e-110 or -9.999999999999969e-311 < J < 5.79999999999999953e-243

   1. Initial program 43.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 43.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 43.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 43.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 65.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 65.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 65.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 37.8%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 37.8%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 37.8%

      \[\leadsto \color{blue}{-U} \]

   if 2.35000000000000017e-229 < J < 5.59999999999999988e-120

   1. Initial program 50.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 50.7%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 50.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 67.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 67.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 67.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 6.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 6.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 6.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 6.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 45.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 45.1%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 45.1%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 45.1%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 45.1%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]

   if 1.59999999999999998e-91 < J < 1.9500000000000001e25

   1. Initial program 57.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 57.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 57.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 57.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 90.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 48.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 48.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 48.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 48.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 48.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 32.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 32.1%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 32.1%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 32.1%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 32.1%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 32.1%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 32.1%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 32.1%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 32.1%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 32.1%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -5.2 \cdot 10^{+16}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.5 \cdot 10^{-90}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -8.8 \cdot 10^{-110}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.35 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.6 \cdot 10^{-120}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{-91} \lor \neg \left(J \leq 1.95 \cdot 10^{+25}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 62.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;K \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;K \leq 6.2 \cdot 10^{+213} \lor \neg \left(K \leq 1.22 \cdot 10^{+240}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= K 8.5e-5)
   (* -2.0 (* J (hypot 1.0 (/ (* U 0.5) J))))
   (if (or (<= K 6.2e+213) (not (<= K 1.22e+240)))
     (* (* -2.0 J) (cos (* K 0.5)))
     (* -2.0 (+ (* J (/ J U)) (* U 0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (K <= 8.5e-5) {
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	} else if ((K <= 6.2e+213) || !(K <= 1.22e+240)) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (K <= 8.5e-5) {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U * 0.5) / J)));
	} else if ((K <= 6.2e+213) || !(K <= 1.22e+240)) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if K <= 8.5e-5:
		tmp = -2.0 * (J * math.hypot(1.0, ((U * 0.5) / J)))
	elif (K <= 6.2e+213) or not (K <= 1.22e+240):
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	else:
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (K <= 8.5e-5)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U * 0.5) / J))));
	elseif ((K <= 6.2e+213) || !(K <= 1.22e+240))
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (K <= 8.5e-5)
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	elseif ((K <= 6.2e+213) || ~((K <= 1.22e+240)))
		tmp = (-2.0 * J) * cos((K * 0.5));
	else
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[K, 8.5e-5], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[K, 6.2e+213], N[Not[LessEqual[K, 1.22e+240]], $MachinePrecision]], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if K < 8.500000000000001e-5

   1. Initial program 71.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 71.7%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 71.7%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 71.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 87.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 87.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 87.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 37.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 37.3%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \color{blue}{\sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}} \cdot \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}}}\right) \]

      2. hypot-1-def 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{hypot}\left(1, \sqrt{0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)}\right) \]

      3. *-commutative 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{U \cdot U}{J \cdot J} \cdot 0.25}}\right)\right) \]

      4. times-frac 52.4%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\color{blue}{\left(\frac{U}{J} \cdot \frac{U}{J}\right)} \cdot 0.25}\right)\right) \]

   8. Applied egg-rr 52.4%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{hypot}\left(1, \sqrt{\left(\frac{U}{J} \cdot \frac{U}{J}\right) \cdot 0.25}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 52.4%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\color{blue}{0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)}}\right)\right) \]

      2. times-frac 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{0.25 \cdot \color{blue}{\frac{U \cdot U}{J \cdot J}}}\right)\right) \]

      3. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{0.25 \cdot \frac{\color{blue}{{U}^{2}}}{J \cdot J}}\right)\right) \]

      4. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{0.25 \cdot \frac{{U}^{2}}{\color{blue}{{J}^{2}}}}\right)\right) \]

      5. associate-*r/ 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{0.25 \cdot {U}^{2}}{{J}^{2}}}}\right)\right) \]

      6. associate-/l* 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{0.25}{\frac{{J}^{2}}{{U}^{2}}}}}\right)\right) \]

      7. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\frac{\color{blue}{J \cdot J}}{{U}^{2}}}}\right)\right) \]

      8. unpow2 37.3%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\frac{J \cdot J}{\color{blue}{U \cdot U}}}}\right)\right) \]

      9. times-frac 52.4%

         \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\color{blue}{\frac{J}{U} \cdot \frac{J}{U}}}}\right)\right) \]

   10. Simplified 52.4%

       \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\frac{J}{U} \cdot \frac{J}{U}}}\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-log1p-u 33.8%

          \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\frac{J}{U} \cdot \frac{J}{U}}}\right)\right)\right)} \]

       2. expm1-udef 17.7%

          \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \sqrt{\frac{0.25}{\frac{J}{U} \cdot \frac{J}{U}}}\right)\right)} - 1\right)} \]

       3. sqrt-div 18.4%

          \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{\sqrt{0.25}}{\sqrt{\frac{J}{U} \cdot \frac{J}{U}}}}\right)\right)} - 1\right) \]

       4. metadata-eval 18.4%

          \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{0.5}}{\sqrt{\frac{J}{U} \cdot \frac{J}{U}}}\right)\right)} - 1\right) \]

       5. sqrt-prod 11.4%

          \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\color{blue}{\sqrt{\frac{J}{U}} \cdot \sqrt{\frac{J}{U}}}}\right)\right)} - 1\right) \]

       6. add-sqr-sqrt 23.6%

          \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\color{blue}{\frac{J}{U}}}\right)\right)} - 1\right) \]

   12. Applied egg-rr 23.6%

       \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 41.6%

          \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\right)\right)} \]

       2. expm1-log1p 64.9%

          \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\right)} \]

       3. associate-/l* 65.0%

          \[\leadsto -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{0.5 \cdot U}{J}}\right)\right) \]

   14. Simplified 65.0%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \mathsf{hypot}\left(1, \frac{0.5 \cdot U}{J}\right)\right)} \]

   if 8.500000000000001e-5 < K < 6.19999999999999981e213 or 1.2200000000000001e240 < K

   1. Initial program 67.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 67.1%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 67.1%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 67.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 82.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 82.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 82.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 42.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 42.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 42.8%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 42.8%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 42.8%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 42.8%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 42.8%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   if 6.19999999999999981e213 < K < 1.2200000000000001e240

   1. Initial program 35.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 35.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 35.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 35.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.5%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 7.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 7.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 68.4%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 68.4%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 68.4%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 68.4%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 68.4%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 69.3%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 69.3%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 69.3%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;K \leq 6.2 \cdot 10^{+213} \lor \neg \left(K \leq 1.22 \cdot 10^{+240}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 49.4% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{+99}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -3.4e+42)
   (* J (+ -2.0 (* 0.25 (* K K))))
   (if (<= J -1e-310)
     U
     (if (<= J 1.4e-243)
       (- U)
       (if (<= J 2.1e-229)
         U
         (if (<= J 3.6e+99)
           (* -2.0 (+ (* J (/ J U)) (* U 0.5)))
           (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -3.4e+42) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 1.4e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 3.6e+99) {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-3.4d+42)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 1.4d-243) then
        tmp = -u
    else if (j <= 2.1d-229) then
        tmp = u
    else if (j <= 3.6d+99) then
        tmp = (-2.0d0) * ((j * (j / u)) + (u * 0.5d0))
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -3.4e+42) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 1.4e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 3.6e+99) {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -3.4e+42:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -1e-310:
		tmp = U
	elif J <= 1.4e-243:
		tmp = -U
	elif J <= 2.1e-229:
		tmp = U
	elif J <= 3.6e+99:
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5))
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -3.4e+42)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 1.4e-243)
		tmp = Float64(-U);
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 3.6e+99)
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -3.4e+42)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 1.4e-243)
		tmp = -U;
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 3.6e+99)
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -3.4e+42], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -1e-310], U, If[LessEqual[J, 1.4e-243], (-U), If[LessEqual[J, 2.1e-229], U, If[LessEqual[J, 3.6e+99], N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -3.39999999999999975e42

   1. Initial program 95.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 95.2%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 95.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 75.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 75.0%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 75.0%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 75.0%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 75.0%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   7. Taylor expanded in K around 0 45.7%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 45.7%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 45.7%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 45.7%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 45.7%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 45.7%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -3.39999999999999975e42 < J < -9.999999999999969e-311 or 1.39999999999999997e-243 < J < 2.09999999999999983e-229

   1. Initial program 44.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 44.4%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 44.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 40.5%

      \[\leadsto \color{blue}{U} \]

   if -9.999999999999969e-311 < J < 1.39999999999999997e-243

   1. Initial program 32.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 32.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 32.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 32.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 42.6%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 42.6%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 42.6%

      \[\leadsto \color{blue}{-U} \]

   if 2.09999999999999983e-229 < J < 3.6000000000000002e99

   1. Initial program 68.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 68.1%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 68.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 35.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 35.8%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 35.8%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 35.8%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 35.8%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 28.3%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 28.3%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 28.3%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 28.3%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 28.3%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 28.3%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 28.3%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 28.3%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 28.3%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 28.3%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]

   if 3.6000000000000002e99 < J

   1. Initial program 99.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 99.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 99.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around inf 41.4%

      \[\leadsto -2 \cdot \color{blue}{J} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{+99}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9: 49.5% accurate, 19.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.2 \cdot 10^{+35}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.7 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + \frac{U}{\frac{J}{U}} \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.2e+35)
   (* J (+ -2.0 (* 0.25 (* K K))))
   (if (<= J -1e-310)
     U
     (if (<= J 5.8e-243)
       (- U)
       (if (<= J 2.1e-229)
         U
         (if (<= J 1.7e+101)
           (* -2.0 (+ (* J (/ J U)) (* U 0.5)))
           (* -2.0 (+ J (* (/ U (/ J U)) 0.125)))))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.2e+35) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 1.7e+101) {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	} else {
		tmp = -2.0 * (J + ((U / (J / U)) * 0.125));
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.2d+35)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 5.8d-243) then
        tmp = -u
    else if (j <= 2.1d-229) then
        tmp = u
    else if (j <= 1.7d+101) then
        tmp = (-2.0d0) * ((j * (j / u)) + (u * 0.5d0))
    else
        tmp = (-2.0d0) * (j + ((u / (j / u)) * 0.125d0))
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.2e+35) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else if (J <= 1.7e+101) {
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	} else {
		tmp = -2.0 * (J + ((U / (J / U)) * 0.125));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.2e+35:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -1e-310:
		tmp = U
	elif J <= 5.8e-243:
		tmp = -U
	elif J <= 2.1e-229:
		tmp = U
	elif J <= 1.7e+101:
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5))
	else:
		tmp = -2.0 * (J + ((U / (J / U)) * 0.125))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.2e+35)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = Float64(-U);
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 1.7e+101)
		tmp = Float64(-2.0 * Float64(Float64(J * Float64(J / U)) + Float64(U * 0.5)));
	else
		tmp = Float64(-2.0 * Float64(J + Float64(Float64(U / Float64(J / U)) * 0.125)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.2e+35)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = -U;
	elseif (J <= 2.1e-229)
		tmp = U;
	elseif (J <= 1.7e+101)
		tmp = -2.0 * ((J * (J / U)) + (U * 0.5));
	else
		tmp = -2.0 * (J + ((U / (J / U)) * 0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.2e+35], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -1e-310], U, If[LessEqual[J, 5.8e-243], (-U), If[LessEqual[J, 2.1e-229], U, If[LessEqual[J, 1.7e+101], N[(-2.0 * N[(N[(J * N[(J / U), $MachinePrecision]), $MachinePrecision] + N[(U * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(J + N[(N[(U / N[(J / U), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if J < -2.1999999999999999e35

   1. Initial program 95.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 95.2%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 95.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 75.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 75.0%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 75.0%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 75.0%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 75.0%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   7. Taylor expanded in K around 0 45.7%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 45.7%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 45.7%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 45.7%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 45.7%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 45.7%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -2.1999999999999999e35 < J < -9.999999999999969e-311 or 5.79999999999999953e-243 < J < 2.09999999999999983e-229

   1. Initial program 44.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 44.4%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 44.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 40.5%

      \[\leadsto \color{blue}{U} \]

   if -9.999999999999969e-311 < J < 5.79999999999999953e-243

   1. Initial program 32.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 32.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 32.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 32.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 42.6%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 42.6%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 42.6%

      \[\leadsto \color{blue}{-U} \]

   if 2.09999999999999983e-229 < J < 1.70000000000000009e101

   1. Initial program 68.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 68.1%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 68.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 84.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 35.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 35.8%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 35.8%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 35.8%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 35.8%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around 0 28.3%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U + \frac{{J}^{2}}{U}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 28.3%

         \[\leadsto -2 \cdot \left(\color{blue}{U \cdot 0.5} + \frac{{J}^{2}}{U}\right) \]

      2. fma-def 28.3%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{{J}^{2}}{U}\right)} \]

      3. unpow2 28.3%

         \[\leadsto -2 \cdot \mathsf{fma}\left(U, 0.5, \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 28.3%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)} \]
   10. Step-by-step derivation

       1. fma-udef 28.3%

          \[\leadsto -2 \cdot \color{blue}{\left(U \cdot 0.5 + \frac{J \cdot J}{U}\right)} \]

       2. +-commutative 28.3%

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{J \cdot J}{U} + U \cdot 0.5\right)} \]

       3. associate-*r/ 28.3%

          \[\leadsto -2 \cdot \left(\color{blue}{J \cdot \frac{J}{U}} + U \cdot 0.5\right) \]

       4. *-commutative 28.3%

          \[\leadsto -2 \cdot \left(J \cdot \frac{J}{U} + \color{blue}{0.5 \cdot U}\right) \]

   11. Applied egg-rr 28.3%

       \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \frac{J}{U} + 0.5 \cdot U\right)} \]

   if 1.70000000000000009e101 < J

   1. Initial program 99.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 99.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 99.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around inf 40.7%

      \[\leadsto -2 \cdot \color{blue}{\left(0.125 \cdot \frac{{U}^{2}}{J} + J\right)} \]
   8. Step-by-step derivation

      1. *-commutative 40.7%

         \[\leadsto -2 \cdot \left(\color{blue}{\frac{{U}^{2}}{J} \cdot 0.125} + J\right) \]

      2. fma-def 40.7%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(\frac{{U}^{2}}{J}, 0.125, J\right)} \]

      3. unpow2 40.7%

         \[\leadsto -2 \cdot \mathsf{fma}\left(\frac{\color{blue}{U \cdot U}}{J}, 0.125, J\right) \]

      4. associate-*r/ 41.4%

         \[\leadsto -2 \cdot \mathsf{fma}\left(\color{blue}{U \cdot \frac{U}{J}}, 0.125, J\right) \]

   9. Simplified 41.4%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(U \cdot \frac{U}{J}, 0.125, J\right)} \]
   10. Step-by-step derivation

       1. fma-udef 41.4%

          \[\leadsto -2 \cdot \color{blue}{\left(\left(U \cdot \frac{U}{J}\right) \cdot 0.125 + J\right)} \]

       2. clear-num 41.4%

          \[\leadsto -2 \cdot \left(\left(U \cdot \color{blue}{\frac{1}{\frac{J}{U}}}\right) \cdot 0.125 + J\right) \]

       3. un-div-inv 41.4%

          \[\leadsto -2 \cdot \left(\color{blue}{\frac{U}{\frac{J}{U}}} \cdot 0.125 + J\right) \]

   11. Applied egg-rr 41.4%

       \[\leadsto -2 \cdot \color{blue}{\left(\frac{U}{\frac{J}{U}} \cdot 0.125 + J\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.2 \cdot 10^{+35}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.7 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U} + U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J + \frac{U}{\frac{J}{U}} \cdot 0.125\right)\\ \end{array} \]

Alternative 10: 50.2% accurate, 31.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.7 \cdot 10^{+99}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -2.1e+64)
   (* -2.0 J)
   (if (<= J -1e-310)
     U
     (if (<= J 5.8e-243)
       (- U)
       (if (<= J 3.1e-229) U (if (<= J 3.7e+99) (- U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.1e+64) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 3.1e-229) {
		tmp = U;
	} else if (J <= 3.7e+99) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.1d+64)) then
        tmp = (-2.0d0) * j
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 5.8d-243) then
        tmp = -u
    else if (j <= 3.1d-229) then
        tmp = u
    else if (j <= 3.7d+99) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2.1e+64) {
		tmp = -2.0 * J;
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 3.1e-229) {
		tmp = U;
	} else if (J <= 3.7e+99) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2.1e+64:
		tmp = -2.0 * J
	elif J <= -1e-310:
		tmp = U
	elif J <= 5.8e-243:
		tmp = -U
	elif J <= 3.1e-229:
		tmp = U
	elif J <= 3.7e+99:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2.1e+64)
		tmp = Float64(-2.0 * J);
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = Float64(-U);
	elseif (J <= 3.1e-229)
		tmp = U;
	elseif (J <= 3.7e+99)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2.1e+64)
		tmp = -2.0 * J;
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = -U;
	elseif (J <= 3.1e-229)
		tmp = U;
	elseif (J <= 3.7e+99)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2.1e+64], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -1e-310], U, If[LessEqual[J, 5.8e-243], (-U), If[LessEqual[J, 3.1e-229], U, If[LessEqual[J, 3.7e+99], (-U), N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.1e64 or 3.7000000000000001e99 < J

   1. Initial program 97.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 97.8%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 97.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 97.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 41.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 41.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 41.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 41.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 41.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around inf 46.0%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -2.1e64 < J < -9.999999999999969e-311 or 5.79999999999999953e-243 < J < 3.1000000000000001e-229

   1. Initial program 48.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 48.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 48.8%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 77.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 77.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 77.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 39.2%

      \[\leadsto \color{blue}{U} \]

   if -9.999999999999969e-311 < J < 5.79999999999999953e-243 or 3.1000000000000001e-229 < J < 3.7000000000000001e99

   1. Initial program 61.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 61.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 61.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 61.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 31.2%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 31.2%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 31.2%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.7 \cdot 10^{+99}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 11: 49.5% accurate, 31.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{+101}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -4.8e+40)
   (* J (+ -2.0 (* 0.25 (* K K))))
   (if (<= J -1e-310)
     U
     (if (<= J 4.6e-243)
       (- U)
       (if (<= J 2.2e-229) U (if (<= J 1.9e+101) (- U) (* -2.0 J)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -4.8e+40) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 4.6e-243) {
		tmp = -U;
	} else if (J <= 2.2e-229) {
		tmp = U;
	} else if (J <= 1.9e+101) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-4.8d+40)) then
        tmp = j * ((-2.0d0) + (0.25d0 * (k * k)))
    else if (j <= (-1d-310)) then
        tmp = u
    else if (j <= 4.6d-243) then
        tmp = -u
    else if (j <= 2.2d-229) then
        tmp = u
    else if (j <= 1.9d+101) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -4.8e+40) {
		tmp = J * (-2.0 + (0.25 * (K * K)));
	} else if (J <= -1e-310) {
		tmp = U;
	} else if (J <= 4.6e-243) {
		tmp = -U;
	} else if (J <= 2.2e-229) {
		tmp = U;
	} else if (J <= 1.9e+101) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -4.8e+40:
		tmp = J * (-2.0 + (0.25 * (K * K)))
	elif J <= -1e-310:
		tmp = U
	elif J <= 4.6e-243:
		tmp = -U
	elif J <= 2.2e-229:
		tmp = U
	elif J <= 1.9e+101:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -4.8e+40)
		tmp = Float64(J * Float64(-2.0 + Float64(0.25 * Float64(K * K))));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 4.6e-243)
		tmp = Float64(-U);
	elseif (J <= 2.2e-229)
		tmp = U;
	elseif (J <= 1.9e+101)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -4.8e+40)
		tmp = J * (-2.0 + (0.25 * (K * K)));
	elseif (J <= -1e-310)
		tmp = U;
	elseif (J <= 4.6e-243)
		tmp = -U;
	elseif (J <= 2.2e-229)
		tmp = U;
	elseif (J <= 1.9e+101)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -4.8e+40], N[(J * N[(-2.0 + N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, -1e-310], U, If[LessEqual[J, 4.6e-243], (-U), If[LessEqual[J, 2.2e-229], U, If[LessEqual[J, 1.9e+101], (-U), N[(-2.0 * J), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -4.8e40

   1. Initial program 95.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 95.2%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 95.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.9%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around inf 75.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J} \]

      2. *-commutative 75.0%

         \[\leadsto \left(-2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J \]

      3. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J \]

      4. associate-*r* 75.0%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)} \]

      5. *-commutative 75.0%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]

   6. Simplified 75.0%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   7. Taylor expanded in K around 0 45.7%

      \[\leadsto \color{blue}{0.25 \cdot \left({K}^{2} \cdot J\right) + -2 \cdot J} \]
   8. Step-by-step derivation

      1. +-commutative 45.7%

         \[\leadsto \color{blue}{-2 \cdot J + 0.25 \cdot \left({K}^{2} \cdot J\right)} \]

      2. associate-*r* 45.7%

         \[\leadsto -2 \cdot J + \color{blue}{\left(0.25 \cdot {K}^{2}\right) \cdot J} \]

      3. distribute-rgt-out 45.7%

         \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot {K}^{2}\right)} \]

      4. unpow2 45.7%

         \[\leadsto J \cdot \left(-2 + 0.25 \cdot \color{blue}{\left(K \cdot K\right)}\right) \]

   9. Simplified 45.7%

      \[\leadsto \color{blue}{J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)} \]

   if -4.8e40 < J < -9.999999999999969e-311 or 4.6e-243 < J < 2.1999999999999999e-229

   1. Initial program 44.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 44.4%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 44.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 75.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 40.5%

      \[\leadsto \color{blue}{U} \]

   if -9.999999999999969e-311 < J < 4.6e-243 or 2.1999999999999999e-229 < J < 1.8999999999999999e101

   1. Initial program 61.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 61.0%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 61.0%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 61.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 80.0%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 31.2%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 31.2%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 31.2%

      \[\leadsto \color{blue}{-U} \]

   if 1.8999999999999999e101 < J

   1. Initial program 99.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 99.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 99.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 99.7%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in K around 0 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{\color{blue}{U \cdot U}}{{J}^{2}}}\right) \]

      3. unpow2 39.9%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{\color{blue}{J \cdot J}}}\right) \]

   6. Simplified 39.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)} \]

   7. Taylor expanded in J around inf 41.4%

      \[\leadsto -2 \cdot \color{blue}{J} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.8 \cdot 10^{+40}:\\ \;\;\;\;J \cdot \left(-2 + 0.25 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{+101}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 12: 39.1% accurate, 51.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq 10^{-308}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J 1e-308) U (if (<= J 5.8e-243) (- U) (if (<= J 2.1e-229) U (- U)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= 1e-308) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= 1d-308) then
        tmp = u
    else if (j <= 5.8d-243) then
        tmp = -u
    else if (j <= 2.1d-229) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= 1e-308) {
		tmp = U;
	} else if (J <= 5.8e-243) {
		tmp = -U;
	} else if (J <= 2.1e-229) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= 1e-308:
		tmp = U
	elif J <= 5.8e-243:
		tmp = -U
	elif J <= 2.1e-229:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= 1e-308)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = Float64(-U);
	elseif (J <= 2.1e-229)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= 1e-308)
		tmp = U;
	elseif (J <= 5.8e-243)
		tmp = -U;
	elseif (J <= 2.1e-229)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, 1e-308], U, If[LessEqual[J, 5.8e-243], (-U), If[LessEqual[J, 2.1e-229], U, (-U)]]]

Derivation

1. Split input into 2 regimes

2. if J < 9.9999999999999991e-309 or 5.79999999999999953e-243 < J < 2.09999999999999983e-229

   1. Initial program 67.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 67.1%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 67.1%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 67.1%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 86.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 86.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 86.2%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in U around -inf 28.1%

      \[\leadsto \color{blue}{U} \]

   if 9.9999999999999991e-309 < J < 5.79999999999999953e-243 or 2.09999999999999983e-229 < J

   1. Initial program 73.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. *-commutative 73.6%

         \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 73.6%

         \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. unpow2 73.6%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

      4. hypot-1-def 86.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      5. *-commutative 86.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      6. associate-*l* 86.4%

         \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

   4. Taylor expanded in J around 0 23.8%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   5. Step-by-step derivation

      1. neg-mul-1 23.8%

         \[\leadsto \color{blue}{-U} \]

   6. Simplified 23.8%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 10^{-308}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-229}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 13: 26.6% accurate, 420.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ U \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 U)

C

U = abs(U);
double code(double J, double K, double U) {
	return U;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	return U;
}

Python

U = abs(U)
def code(J, K, U):
	return U

Julia

U = abs(U)
function code(J, K, U)
	return U
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	tmp = U;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := U

Derivation

1. Initial program 70.3%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Step-by-step derivation

   1. *-commutative 70.3%

      \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. associate-*l* 70.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. unpow2 70.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]

   4. hypot-1-def 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   5. *-commutative 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\left(J \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   6. associate-*l* 86.3%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Simplified 86.3%

   \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]

4. Taylor expanded in U around -inf 30.3%

   \[\leadsto \color{blue}{U} \]

5. Final simplification 30.3%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023192 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))